Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $z = \dfrac{-p + 8}{p^2 - 15p + 50} \div \dfrac{9p - 72}{-7p^2 + 70p} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{-p + 8}{p^2 - 15p + 50} \times \dfrac{-7p^2 + 70p}{9p - 72} $ First factor the quadratic. $z = \dfrac{-p + 8}{(p - 10)(p - 5)} \times \dfrac{-7p^2 + 70p}{9p - 72} $ Then factor out any other terms. $z = \dfrac{-(p - 8)}{(p - 10)(p - 5)} \times \dfrac{-7p(p - 10)}{9(p - 8)} $ Then multiply the two numerators and multiply the two denominators. $z = \dfrac{ -(p - 8) \times -7p(p - 10) } { (p - 10)(p - 5) \times 9(p - 8) } $ $z = \dfrac{ 7p(p - 8)(p - 10)}{ 9(p - 10)(p - 5)(p - 8)} $ Notice that $(p - 8)$ and $(p - 10)$ appear in both the numerator and denominator so we can cancel them. $z = \dfrac{ 7p(p - 8)\cancel{(p - 10)}}{ 9\cancel{(p - 10)}(p - 5)(p - 8)} $ We are dividing by $p - 10$ , so $p - 10 \neq 0$ Therefore, $p \neq 10$ $z = \dfrac{ 7p\cancel{(p - 8)}\cancel{(p - 10)}}{ 9\cancel{(p - 10)}(p - 5)\cancel{(p - 8)}} $ We are dividing by $p - 8$ , so $p - 8 \neq 0$ Therefore, $p \neq 8$ $z = \dfrac{7p}{9(p - 5)} ; \space p \neq 10 ; \space p \neq 8 $